US20070244945A1 - Analog circuit arrangement for creating elliptic functions - Google Patents
Analog circuit arrangement for creating elliptic functions Download PDFInfo
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- G06G7/24—Arrangements for performing computing operations, e.g. operational amplifiers for evaluating logarithmic or exponential functions, e.g. hyperbolic functions
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- G06G7/32—Arrangements for performing computing operations, e.g. operational amplifiers for solving of equations or inequations; for matrices
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Definitions
- the present invention relates to an analog circuit system having a plurality of analog computing circuits for generating elliptic functions.
- Elliptic functions and integrals are used in numerous applications in engineering practice.
- the elliptic functions occurring frequently are the so-called Jacobi elliptic functions sn(x,k), cn(x,k), dn(x,k).
- the characteristic of the function sn(x,k) is similar to the sine function, while the function cn(x,k) is similar to the cosine function.
- the value of k lies mostly in the interval [0, 1].
- Elliptic functions play a role in information and communication technology, e.g., in the design of Cauer filters, because some parameters of the Cauer filter are linked by elliptic functions.
- German patent reference 102 49 050.3 apparently describes a method and an arrangement for 20- adjusting an analog filter with the aid of elliptic functions.
- Elliptic functions are likewise used in the two-dimensional representation, interpolation or compression of data, for example, see German patent reference 102 48 543.7.
- the present invention provides for analog circuit systems that are able to electrically simulate elliptic functions.
- an analog circuit system has a plurality of analog computing circuits such as analog multipliers, adders, integrators, differential amplifiers and dividers, which generate at least one output signal whose curve shape, at least sectionally, corresponds or is approximate to an elliptic function.
- analog computing circuits such as analog multipliers, adders, integrators, differential amplifiers and dividers, which generate at least one output signal whose curve shape, at least sectionally, corresponds or is approximate to an elliptic function.
- Jacobi elliptic functions are electrically simulated by the analog circuit system.
- an analog circuit system includes analog multipliers and integrators which are able to deliver three output signals whose curve shapes, at least sectionally, correspond or are approximate to the Jacobi elliptic time functions sn ( 2 ⁇ ⁇ ⁇ ⁇ T ⁇ t , k ) , cn ( 2 ⁇ ⁇ ⁇ T ⁇ t , k ) ⁇ ⁇ and ⁇ ⁇ dn ( 2 ⁇ ⁇ ⁇ ⁇ T ⁇ t , k ) .
- k is the module of the elliptic functions
- ⁇ ⁇ ⁇ M ( 1 , 1 - k 2 )
- M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ) represents the so-called arithmetic-geometric mean of 1 and ⁇ square root over (1 ⁇ k 2 ) ⁇ .
- the value k lies mostly in the interval [0, 1].
- a plurality of analog computing circuits are interconnected in such a way that, given an input variable x, output variable y is an elliptic function of x.
- a circuit system able to generate this functional relationship has a first multiplier, at whose one input an input signal having the quantity x, for example, a triangular input signal, is applied, and at whose other input the factor (1 ⁇ k 2 )/2 is applied.
- a second multiplier can be provided, at whose one input the triangular input signal is applied, and at whose other input the factor (1+k 2 )/2 is applied.
- a differential amplifier is connected to the output of the second multiplier, a further input of the differential amplifier being connected to ground.
- An adder is also provided which is connected to the output of the first multiplier and the output of the differential amplifier. Present at the output of the adder is an output signal U a which is combined or linked with the input signal by the Jacobi elliptic function sn(U e ).
- output signals cn ( 2 ⁇ ⁇ ⁇ T ⁇ t , k ) ⁇ ⁇ and ⁇ ⁇ dn ( 2 ⁇ ⁇ ⁇ ⁇ T ⁇ t , k ) are applied to the inputs of the analog division device.
- At least one analog computing circuit is provided, at whose first input, the value 1 is applied, and at whose second input, the factor ⁇ square root over (1 ⁇ k 2 ) ⁇ is applied.
- the arithmetic mean of the two input signals is present at the first output of the analog computing circuit, whereas the geometric mean of the two input signals is present at the second output of the analog computing circuit.
- an analog computing circuit connected to the outputs of the analog computing devices or circuits, is provided for calculating the arithmetic mean, which corresponds approximately to the arithmetic-geometric mean M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ) of 1 and ⁇ square root over (1 ⁇ k 2 ) ⁇ .
- An alternative analog circuit system for generating the arithmetic-geometric mean M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ) has one analog computing circuit for calculating the minimum from two input signals, one analog computing circuit for calculating the maximum from two input signals, one analog computing circuit for calculating the arithmetic mean from two input signals, and one analog computing circuit for calculating the geometric mean from two input signals.
- the output of the analog computing circuit for calculating the minimum is connected to an input of the analog computing circuit for calculating the arithmetic mean and an input of the analog computing circuit for calculating the geometric mean.
- the output of the analog computing circuit for calculating the maximum is connected to another input of the analog computing circuit for calculating the arithmetic mean and another input of the analog computing circuit for calculating the geometric mean.
- One input of the analog computing circuit for calculating the minimum is connected to the output of the analog computing circuit for calculating the arithmetic mean, the value 1 being applied to the other input.
- One input of the analog computing circuit for calculating the maximum is connected to the output of the analog computing circuit for calculating the geometric mean, the value ⁇ square root over (1 ⁇ k 2 ) ⁇ being applied to the other input.
- the arithmetic-geometric mean M o f 1 and ⁇ square root over (1 ⁇ k 2 ) ⁇ is present at the output of the analog computing circuit for calculating the geometric mean and at the output of the analog computing circuit for calculating the arithmetic mean.
- a device for example, a divider, is provided, at whose inputs, the arithmetic-geometric mean M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ) and the number ⁇ are applied.
- FIG. 1 shows an analog circuit system for generating three output signals, each corresponding to a Jacobi elliptic time function.
- FIG. 2 shows an analog circuit system for generating an output signal which corresponds to the Jacobi elliptic time function sn ( 2 ⁇ ⁇ ⁇ ⁇ T ⁇ t ) .
- FIG. 3 shows an analog circuit system for generating an output signal which is combined with a triangular input signal by the Jacobi elliptic time function sn(U e ).
- FIG. 4 shows an analog circuit system which, from two input signals, supplies an estimate for the arithmetic-geometric mean M.
- FIG. 5 shows an alternative analog circuit system for calculating the arithmetic-geometric mean M from two input signals.
- FIG. 6 shows a divider for generating the value ⁇ circumflex over ( ⁇ ) ⁇ .
- analog circuit systems which generate at least one output signal whose curve shape corresponds or is approximate to a Jacobi elliptic time function.
- Jacobi elliptic functions sn(x,k), cn(x,k) and dn(x,k) are used in the following embodiment.
- the variable x is replaced by t in the above functions, and, to simplify matters, the value of k is omitted in the following formulas.
- variable ⁇ ⁇ M ⁇ ( 1 , 1 - k 2 ) ( 4 )
- the function M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ) forms the so-called arithmetic-geometric mean of 1 and ( ⁇ square root over (1 ⁇ k 2 ) ⁇ ).
- FIG. 1 shows an analog circuit system which generates three output signals whose curve shapes correspond to the Jacobi elliptic functions.
- a multiplier 10 , a multiplier 20 , and an analog integrator 30 are connected in series.
- an analog multiplier 40 , an analog multiplier 50 , and a further analog integrator 60 are connected in series.
- a third series connection includes a further analog multiplier 70 , an analog multiplier 80 , and an analog integrator 90 .
- Analog multiplier 20 multiplies the output signal of multiplier 10 by the factor 2 ⁇ circumflex over ( ⁇ ) ⁇ /T.
- Multiplier 50 multiplies the output signal of multiplier 40 by the factor - 2 ⁇ ⁇ ⁇ ⁇ T .
- Multiplier 80 multiplies the output signal of multiplier 70 by the factor - k 2 ⁇ 2 ⁇ ⁇ ⁇ ⁇ T .
- the output signal of integrator 30 is coupled back to multiplier 40 and to the input of multiplier 70 .
- the output signal of integrator 60 is coupled back to the input of multiplier 10 and to the input of multiplier 70 .
- the output of integrator 90 is coupled back to the input of multiplier 40 and to the input of multiplier 10 .
- the multiplication by ⁇ 2 ⁇ ⁇ ⁇ ⁇ T in multipliers 20 , 50 , respectively, and the multiplication by - k 2 ⁇ 2 ⁇ ⁇ ⁇ ⁇ T in multiplier 80 may also be carried out in integrators 30 , 60 , 90 .
- the multiplication by k 2 may also be put at the output of integrator 90 .
- All three Jacobi elliptic time functions sn(2 ⁇ circumflex over ( ⁇ ) ⁇ ft), cn(2 ⁇ circumflex over ( ⁇ ) ⁇ ft) and dn(2 ⁇ circumflex over ( ⁇ ) ⁇ ft) may be realized simultaneously using the analog circuit system shown in FIG. 1 .
- the derivatives of the Jacobi elliptic time functions sn, cn and dn are obtained at the output of the multipliers 10 , 40 , 70 , respectively.
- FIG. 2 An exemplary analog circuit system which simulates this differential equation (8) is shown in FIG. 2 .
- the analog circuit system has a multiplier 100 whose output is connected to a series-connected multiplier 110 . Moreover, the factor ⁇ 2k 2 is applied to the input of multiplier 110 . The output of multiplier 110 is connected to an input of an adder 120 . The factor 1+k 2 is applied to a second input of adder 120 . The output of adder 120 is connected to the input of a multiplier 130 . The factor - ( 2 ⁇ ⁇ ⁇ ⁇ T ) 2 is applied to a further input of multiplier 130 . The output of multiplier 130 is connected to an input of a multiplier 140 . The output of multiplier 140 is connected to an input of an integrator 150 . The output of integrator 150 is connected to the input of an integrator 160 .
- integrator 160 The output of integrator 160 is coupled back to the input of multiplier 140 and to two inputs of multiplier 100 . In this way, an output signal whose curve shape corresponds to the Jacobi elliptic time function s ⁇ ⁇ n ⁇ ( 2 ⁇ ⁇ ⁇ ⁇ T ⁇ t ) appears at the output of integrator 160 .
- the multiplication by the factor ( 2 ⁇ ⁇ ⁇ ⁇ T ) 2 may expediently be carried out again in integrators 150 and 160 .
- FIG. 3 an exemplary embodiment is described in which a functional relationship corresponding to the Jacobi elliptic function sn(2 ⁇ circumflex over ( ⁇ ) ⁇ ft) approximatively exists between an input signal and an output signal.
- the analog circuit system shown in FIG. 3 includes a differential amplifier 170 , a multiplier 180 , a multiplier 190 and an adder 200 .
- An input signal having a triangular voltage curve is applied, for example, at each input of the multipliers 180 , 190 .
- the factor (1 ⁇ k 2 )/2 is applied to multiplier 180
- the factor (1+k 2 )/2 is applied to multiplier 190 .
- the output signal of multiplier 190 is fed to differential amplifier 170 .
- the second input of the differential amplifier is connected to ground.
- the output of multiplier 180 and the output of differential amplifier 170 are connected to the inputs of adder 200 .
- combining or linking an output signal and an input signal via the Jacobi elliptic function cn or dn in a circuit system is available knowledge in the art.
- a division device (not shown) may be connected in series to the circuit system shown in FIG. 1 .
- the output signals of the integrators 30 , 60 may be fed (or added) to the division device.
- Equation (4) it is possible to change the value ⁇ circumflex over ( ⁇ ) ⁇ by changing the value k. That is to say, ⁇ circumflex over ( ⁇ ) ⁇ and therefore k may be calculated by calculating the arithmetic-geometric mean M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ).
- One possibility for altering the frequency of the Jacobi elliptic functions generated using the circuit system according to FIG. 1 is to feed a selectively altered value for ⁇ circumflex over ( ⁇ ) ⁇ to the multipliers 20 , 50 , 80 .
- the arithmetic-geometric mean M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ) may be realized, for example, using an analog circuit system which is shown in FIG. 4 .
- the circuit system shown in FIG. 4 is made up of a plurality of analog computing circuits 210 , 220 , 230 , denoted by AG, as well as an analog computing circuit 240 for calculating the arithmetic mean from two input signals.
- Some analog computing circuits 210 , 220 , 230 are adapted in such a way that they generate the arithmetic mean of the two input signals at one output, and the geometric mean of the two input signals at the other output.
- the factor 1 is applied to the first input of analog computing circuit 210
- the factor ⁇ square root over (1 ⁇ k 2 ) ⁇ is applied to its other input.
- the output signal of analog computing circuit 240 corresponds approximately to the arithmetic-geometric mean M of the factors 1 and ⁇ square root over (1 ⁇ k 2 ) ⁇ applied to the inputs of analog computing circuit 210 .
- FIG. 5 shows an alternative analog circuit system for calculating the arithmetic-geometric mean M of the two factors 1 and ⁇ square root over (1 ⁇ k 2 ) ⁇ .
- the circuit system shown in FIG. 5 has an analog computing circuit 250 for calculating the minimum from two input signals, an analog computing circuit 260 for calculating the maximum from two input signals, an analog computing circuit 270 for calculating the arithmetic mean from two input signals and an analog computing circuit 280 for calculating a geometric mean from two input signals.
- the factor 1 is applied to an input of analog computing circuit 250
- the factor ⁇ square root over (1 ⁇ k 2 ) ⁇ is applied to an input of analog computing circuit 260 .
- the output of analog computing circuit 250 for calculating the minimum from two input signals is connected to the input of analog computing circuit 270 and analog computing circuit 280 .
- the output of analog computing circuit 260 for calculating the maximum from two input signals is connected to an input of analog computing circuit 270 and an input of analog computing circuit 280 .
- the output of analog computing circuit 270 is connected to an input of analog computing circuit 250
- the output of analog computing circuit 280 is connected to an input of analog computing circuit 260 .
- the outputs of analog computing circuits 270 and 280 in each case supply the arithmetic-geometric mean M of 1 and ⁇ square root over (1 ⁇ k 2 ) ⁇ .
- Transit-time effects which can be handled with methods (e.g., sample-and-hold elements) generally used in circuit engineering, are not taken into account in the technical implementation of the circuit system according to FIG. 5 .
- ⁇ circumflex over ( ⁇ ) ⁇ i may be calculated via a division device 290 , shown in FIG. 6 , at whose inputs are applied the number ⁇ and the arithmetic-geometric mean M(1, ⁇ square root over (1 ⁇ k 2 ) ⁇ ), which is generated, for example, by the circuit shown in FIG. 4 or in FIG. 5 .
Abstract
Description
- The present invention relates to an analog circuit system having a plurality of analog computing circuits for generating elliptic functions.
- Elliptic functions and integrals are used in numerous applications in engineering practice. The elliptic functions occurring frequently are the so-called Jacobi elliptic functions sn(x,k), cn(x,k), dn(x,k). The characteristic of the function sn(x,k) is similar to the sine function, while the function cn(x,k) is similar to the cosine function. For k=0, the functions sn(x,0) and cn(x,0) change into the sine function and cosine function, respectively. The value of k lies mostly in the interval [0, 1].
- Elliptic functions play a role in information and communication technology, e.g., in the design of Cauer filters, because some parameters of the Cauer filter are linked by elliptic functions. German patent reference 102 49 050.3 apparently describes a method and an arrangement for 20- adjusting an analog filter with the aid of elliptic functions.
- Elliptic functions are likewise used in the two-dimensional representation, interpolation or compression of data, for example, see German patent reference 102 48 543.7.
- The present invention provides for analog circuit systems that are able to electrically simulate elliptic functions.
- For example, an analog circuit system has a plurality of analog computing circuits such as analog multipliers, adders, integrators, differential amplifiers and dividers, which generate at least one output signal whose curve shape, at least sectionally, corresponds or is approximate to an elliptic function.
- In embodiments of the present invention, Jacobi elliptic functions are electrically simulated by the analog circuit system.
- In embodiments of the present invention, an analog circuit system includes analog multipliers and integrators which are able to deliver three output signals whose curve shapes, at least sectionally, correspond or are approximate to the Jacobi elliptic time functions
In these time functions, k is the module of the elliptic functions, f=1/T is the frequency of the elliptic time functions, and
where M(1, √{square root over (1−k2)}) represents the so-called arithmetic-geometric mean of 1 and √{square root over (1−k2)}. The value k lies mostly in the interval [0, 1]. - An application case can frequently occur in which a specific output signal is assigned to an input signal. Therefore, in embodiments of the present invention, a plurality of analog computing circuits are interconnected in such a way that, given an input variable x, output variable y is an elliptic function of x.
- If a triangle function is applied as input signal to a circuit system, which, for example, realizes sn(x), an elliptic time function is obtained at the output.
- A circuit system able to generate this functional relationship has a first multiplier, at whose one input an input signal having the quantity x, for example, a triangular input signal, is applied, and at whose other input the factor (1−k2)/2 is applied. A second multiplier can be provided, at whose one input the triangular input signal is applied, and at whose other input the factor (1+k2)/2 is applied. A differential amplifier is connected to the output of the second multiplier, a further input of the differential amplifier being connected to ground. An adder is also provided which is connected to the output of the first multiplier and the output of the differential amplifier. Present at the output of the adder is an output signal Ua which is combined or linked with the input signal by the Jacobi elliptic function sn(Ue).
- Further elliptic functions may be realized with the aid of an analog division device. To generate an output signal according to the elliptic function
output signals
are applied to the analog division device. To generate an output signal according to the elliptic function
output signals
are applied to the inputs of the analog division device. - In many cases, one wants to selectively control or influence the frequency
as well as the value k of an elliptic function. An exemplary application case is, for example, the voltage-controlled change of frequency f, oscillation period T or module k. For this purpose, one should specifically select the value of, frequency f and the value of {circumflex over (π)}. As mentioned above, the variables {circumflex over (π)} and π can have the following relationship: - For this reason, the arithmetic-geometric mean M(1, √{square root over (1−k2)}) can be simulated with the aid of analog computing circuits.
- In embodiments of the present invention, at least one analog computing circuit is provided, at whose first input, the
value 1 is applied, and at whose second input, the factor √{square root over (1−k2)} is applied. The arithmetic mean of the two input signals is present at the first output of the analog computing circuit, whereas the geometric mean of the two input signals is present at the second output of the analog computing circuit. Moreover, an analog computing circuit, connected to the outputs of the analog computing devices or circuits, is provided for calculating the arithmetic mean, which corresponds approximately to the arithmetic-geometric mean
M(1, √{square root over (1−k2)}) of 1 and √{square root over (1−k2)}. - An alternative analog circuit system for generating the arithmetic-geometric mean M(1, √{square root over (1−k2)}) has one analog computing circuit for calculating the minimum from two input signals, one analog computing circuit for calculating the maximum from two input signals, one analog computing circuit for calculating the arithmetic mean from two input signals, and one analog computing circuit for calculating the geometric mean from two input signals. The output of the analog computing circuit for calculating the minimum is connected to an input of the analog computing circuit for calculating the arithmetic mean and an input of the analog computing circuit for calculating the geometric mean. The output of the analog computing circuit for calculating the maximum is connected to another input of the analog computing circuit for calculating the arithmetic mean and another input of the analog computing circuit for calculating the geometric mean. One input of the analog computing circuit for calculating the minimum is connected to the output of the analog computing circuit for calculating the arithmetic mean, the
value 1 being applied to the other input. One input of the analog computing circuit for calculating the maximum is connected to the output of the analog computing circuit for calculating the geometric mean, the value √{square root over (1−k2)} being applied to the other input. - Consequently, the arithmetic-geometric mean M o f 1 and √{square root over (1−k2)} is present at the output of the analog computing circuit for calculating the geometric mean and at the output of the analog computing circuit for calculating the arithmetic mean.
- To be able to provide the value {circumflex over (π)} in terms of circuit engineering, a device, for example, a divider, is provided, at whose inputs, the arithmetic-geometric mean M(1, √{square root over (1−k2)}) and the number π are applied.
-
FIG. 1 shows an analog circuit system for generating three output signals, each corresponding to a Jacobi elliptic time function. -
FIG. 2 shows an analog circuit system for generating an output signal which corresponds to the Jacobi elliptic time function -
FIG. 3 shows an analog circuit system for generating an output signal which is combined with a triangular input signal by the Jacobi elliptic time function sn(Ue). -
FIG. 4 shows an analog circuit system which, from two input signals, supplies an estimate for the arithmetic-geometric mean M. -
FIG. 5 shows an alternative analog circuit system for calculating the arithmetic-geometric mean M from two input signals. -
FIG. 6 shows a divider for generating the value {circumflex over (π)}. - Herein, analog circuit systems are discussed which generate at least one output signal whose curve shape corresponds or is approximate to a Jacobi elliptic time function. The so-called Jacobi elliptic functions sn(x,k), cn(x,k) and dn(x,k) are used in the following embodiment. In considering time functions, the variable x is replaced by t in the above functions, and, to simplify matters, the value of k is omitted in the following formulas.
- Under these conditions, the following well-known equations may be indicated with respect to the Jacobi elliptic functions:
- Further, descriptions regarding elliptic functions may be found, inter alia, in the reference “Vorlesungen über allgemeine Funktionentheorie und elliptischen Funktionen,” A. Hurwitz, Springer Verlag, 2000, page 204.
- To permit electrical simulation of elliptic functions in which frequency f can be changed, it is necessary, similarly as in the case of the circular functions, to take into account corresponding multiplicative constants which appear in conjunction with variable t. Instead of circular constant π, constant {circumflex over (π)} is used. Variable {circumflex over (π)} has the following relation with variable π:
- The function M(1, √{square root over (1−k2)}) forms the so-called arithmetic-geometric mean of 1 and (√{square root over (1−k2)}).
- With period duration T and the insertion of {circumflex over (π)}, the following differential equations result:
where f=1/T is the frequency of the elliptic functions. -
FIG. 1 shows an analog circuit system which generates three output signals whose curve shapes correspond to the Jacobi elliptic functions. - In
FIG. 1 , amultiplier 10, amultiplier 20, and ananalog integrator 30, are connected in series. Moreover, ananalog multiplier 40, ananalog multiplier 50, and afurther analog integrator 60, are connected in series. A third series connection includes afurther analog multiplier 70, ananalog multiplier 80, and ananalog integrator 90.Analog multiplier 20 multiplies the output signal ofmultiplier 10 by the factor 2 {circumflex over (π)}/T. Multiplier 50 multiplies the output signal ofmultiplier 40 by the factor
Multiplier 80 multiplies the output signal ofmultiplier 70 by the factor - The output signal of
integrator 30 is coupled back tomultiplier 40 and to the input ofmultiplier 70. The output signal ofintegrator 60 is coupled back to the input ofmultiplier 10 and to the input ofmultiplier 70. The output ofintegrator 90 is coupled back to the input ofmultiplier 40 and to the input ofmultiplier 10. Measures, available in circuit engineering, for taking into account predefined initial states during initial operation are not marked in the circuit. Such an analog circuit system, shown inFIG. 1 , delivers the Jacobi elliptic time function sn(2 {circumflex over (π)} ft) at the output ofintegrator 30, the Jacobi elliptic function cn(2 {circumflex over (π)} ft) at the output ofintegrator 60, and the Jacobi elliptic function dn(2 {circumflex over (π)} ft) at the output ofintegrator 90. The multiplication by
inmultipliers
inmultiplier 80 may also be carried out inintegrators integrator 90. Moreover, in further embodiments, it is possible to add familiar stabilization circuits to the circuit system shown inFIG. 1 . See, for example, reference “Halbleiter Schaltungstechnik,” Tietze, Schenk, Springer Verlag, 5th edition, 1980, Berlin, pages 435-438. - All three Jacobi elliptic time functions sn(2 {circumflex over (π)} ft), cn(2 {circumflex over (π)} ft) and dn(2 {circumflex over (π)} ft) may be realized simultaneously using the analog circuit system shown in
FIG. 1 . In addition, the derivatives of the Jacobi elliptic time functions sn, cn and dn are obtained at the output of themultipliers - If, for example, only the Jacobi elliptic time function sn((2 {circumflex over (π)} ft)) is to be realized using an analog circuit system, it is possible to get along with fewer multipliers by considering the differential equation of the second degree, valid for sn(2 {circumflex over (π)} ft), which may be derived from the differential equations indicated above. The differential equation of the second degree valid for sn(2 {circumflex over (π)} ft) reads:
- An exemplary analog circuit system which simulates this differential equation (8) is shown in
FIG. 2 . - The analog circuit system has a
multiplier 100 whose output is connected to a series-connectedmultiplier 110. Moreover, the factor −2k2 is applied to the input ofmultiplier 110. The output ofmultiplier 110 is connected to an input of anadder 120. Thefactor 1+k2 is applied to a second input ofadder 120. The output ofadder 120 is connected to the input of amultiplier 130. The factor
is applied to a further input ofmultiplier 130. The output ofmultiplier 130 is connected to an input of amultiplier 140. The output ofmultiplier 140 is connected to an input of anintegrator 150. The output ofintegrator 150 is connected to the input of anintegrator 160. The output ofintegrator 160 is coupled back to the input ofmultiplier 140 and to two inputs ofmultiplier 100. In this way, an output signal whose curve shape corresponds to the Jacobi elliptic time function
appears at the output ofintegrator 160. - The multiplication by the factor
may expediently be carried out again inintegrators - In
FIG. 3 , an exemplary embodiment is described in which a functional relationship corresponding to the Jacobi elliptic function sn(2 {circumflex over (π)} ft) approximatively exists between an input signal and an output signal. - The analog circuit system shown in
FIG. 3 includes adifferential amplifier 170, amultiplier 180, amultiplier 190 and anadder 200. An input signal having a triangular voltage curve is applied, for example, at each input of themultipliers multiplier 180, whereas the factor (1+k2)/2 is applied tomultiplier 190. The output signal ofmultiplier 190 is fed todifferential amplifier 170. The second input of the differential amplifier is connected to ground. The output ofmultiplier 180 and the output ofdifferential amplifier 170 are connected to the inputs ofadder 200. - Because of the fact that differential-
amplifier circuit 70 has a relation between input signal Ue and output signal Ua according to the equation
given suitably selected parameters of the differential amplifier, the circuit system shown inFIG. 3 generates at the output, a signal Ua, which is approximatively combined with input signal Ue via the Jacobi elliptic function sn. Notably, combining or linking an output signal and an input signal via the Jacobi elliptic function cn or dn in a circuit system is available knowledge in the art. - To be able to generate further elliptic functions, a division device (not shown) may be connected in series to the circuit system shown in
FIG. 1 . For instance, to generate the elliptic function sd(x)=sn(x)/dn(x), the output signals of theintegrators integrators - In embodiments, it may be desirable to selectively control frequency f or the value of k.
- According to equation (4), it is possible to change the value {circumflex over (π)} by changing the value k. That is to say, {circumflex over (π)} and therefore k may be calculated by calculating the arithmetic-geometric mean M(1, √{square root over (1−k2)}). One possibility for altering the frequency of the Jacobi elliptic functions generated using the circuit system according to
FIG. 1 is to feed a selectively altered value for {circumflex over (π)} to themultipliers - To be able to generate {circumflex over (π)} in terms of circuit engineering, the arithmetic-geometric mean M(1, √{square root over (1−k2)}) may be realized, for example, using an analog circuit system which is shown in
FIG. 4 . The circuit system shown inFIG. 4 is made up of a plurality ofanalog computing circuits analog computing circuit 240 for calculating the arithmetic mean from two input signals. Someanalog computing circuits FIG. 4 , thefactor 1 is applied to the first input ofanalog computing circuit 210, and the factor √{square root over (1−k2)} is applied to its other input. On condition that the factor √{square root over (1−k2)} lies between 0 and 1, the output signal ofanalog computing circuit 240 corresponds approximately to the arithmetic-geometric mean M of thefactors 1 and √{square root over (1−k2)} applied to the inputs ofanalog computing circuit 210. -
FIG. 5 shows an alternative analog circuit system for calculating the arithmetic-geometric mean M of the twofactors 1 and √{square root over (1−k2)}. The circuit system shown inFIG. 5 has ananalog computing circuit 250 for calculating the minimum from two input signals, ananalog computing circuit 260 for calculating the maximum from two input signals, ananalog computing circuit 270 for calculating the arithmetic mean from two input signals and ananalog computing circuit 280 for calculating a geometric mean from two input signals. Thefactor 1 is applied to an input ofanalog computing circuit 250, whereas the factor √{square root over (1−k2)} is applied to an input ofanalog computing circuit 260. The output ofanalog computing circuit 250 for calculating the minimum from two input signals is connected to the input ofanalog computing circuit 270 andanalog computing circuit 280. The output ofanalog computing circuit 260 for calculating the maximum from two input signals is connected to an input ofanalog computing circuit 270 and an input ofanalog computing circuit 280. The output ofanalog computing circuit 270 is connected to an input ofanalog computing circuit 250, whereas the output ofanalog computing circuit 280 is connected to an input ofanalog computing circuit 260. In the analog circuit system shown inFIG. 5 , the outputs ofanalog computing circuits - Transit-time effects, which can be handled with methods (e.g., sample-and-hold elements) generally used in circuit engineering, are not taken into account in the technical implementation of the circuit system according to
FIG. 5 . - At this point, {circumflex over (π)} i may be calculated via a
division device 290, shown inFIG. 6 , at whose inputs are applied the number π and the arithmetic-geometric mean M(1, √{square root over (1−k2)}), which is generated, for example, by the circuit shown inFIG. 4 or inFIG. 5 . - In this way, selectively altered values for {circumflex over (π)} may be fed to
multipliers FIG. 1 , which means the frequency response of the output functions may be selectively influenced.
Claims (11)
Applications Claiming Priority (3)
Application Number | Priority Date | Filing Date | Title |
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DE10319637A DE10319637A1 (en) | 2003-05-02 | 2003-05-02 | Analog circuit arrangement for generating elliptical functions |
DE10319637.4 | 2003-05-02 | ||
PCT/DE2004/000223 WO2004097713A2 (en) | 2003-05-02 | 2004-02-09 | Analog circuit arrangement for creating elliptic functions |
Publications (2)
Publication Number | Publication Date |
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US20070244945A1 true US20070244945A1 (en) | 2007-10-18 |
US7584238B2 US7584238B2 (en) | 2009-09-01 |
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US10/555,513 Expired - Fee Related US7584238B2 (en) | 2003-05-02 | 2004-02-09 | Analog circuit system for generating elliptic functions |
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US (1) | US7584238B2 (en) |
EP (1) | EP1623357A2 (en) |
JP (1) | JP4365407B2 (en) |
KR (1) | KR20060119702A (en) |
DE (1) | DE10319637A1 (en) |
WO (1) | WO2004097713A2 (en) |
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CA2900997C (en) * | 2013-03-15 | 2021-03-09 | The Regents Of The University Of California | Fast frequency estimator |
Citations (4)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US3821949A (en) * | 1972-04-10 | 1974-07-02 | Menninger Foundation | Bio-feedback apparatus |
US3900823A (en) * | 1973-03-28 | 1975-08-19 | Nathan O Sokal | Amplifying and processing apparatus for modulated carrier signals |
US5121009A (en) * | 1990-06-15 | 1992-06-09 | Novatel Communications Ltd. | Linear phase low pass filter |
US20040172432A1 (en) * | 2002-10-14 | 2004-09-02 | Klaus Huber | Method for two-dimensional representation, interpolation and compression of data |
Family Cites Families (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
DE2214689A1 (en) | 1972-03-25 | 1973-09-27 | Ver Flugtechnische Werke | CIRCUIT ARRANGEMENT FOR THE FORMATION OF AN OUTPUT SIGNAL FROM SEVERAL INDIVIDUAL SIGNALS |
EP0331246B1 (en) | 1988-02-29 | 1994-01-19 | Koninklijke Philips Electronics N.V. | Logarithmic amplifier |
DE10249050A1 (en) | 2002-10-22 | 2004-05-06 | Huber, Klaus, Dr. | Adjustment of parameters for analog filter of Cauer or elliptical type has circuit producing Cauer word triplets and feeding control signal to filter |
-
2003
- 2003-05-02 DE DE10319637A patent/DE10319637A1/en not_active Withdrawn
-
2004
- 2004-02-09 WO PCT/DE2004/000223 patent/WO2004097713A2/en active Application Filing
- 2004-02-09 KR KR1020057016238A patent/KR20060119702A/en not_active Application Discontinuation
- 2004-02-09 JP JP2006503933A patent/JP4365407B2/en not_active Expired - Fee Related
- 2004-02-09 US US10/555,513 patent/US7584238B2/en not_active Expired - Fee Related
- 2004-02-09 EP EP04709200A patent/EP1623357A2/en not_active Ceased
Patent Citations (4)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US3821949A (en) * | 1972-04-10 | 1974-07-02 | Menninger Foundation | Bio-feedback apparatus |
US3900823A (en) * | 1973-03-28 | 1975-08-19 | Nathan O Sokal | Amplifying and processing apparatus for modulated carrier signals |
US5121009A (en) * | 1990-06-15 | 1992-06-09 | Novatel Communications Ltd. | Linear phase low pass filter |
US20040172432A1 (en) * | 2002-10-14 | 2004-09-02 | Klaus Huber | Method for two-dimensional representation, interpolation and compression of data |
Also Published As
Publication number | Publication date |
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JP4365407B2 (en) | 2009-11-18 |
WO2004097713A2 (en) | 2004-11-11 |
WO2004097713A3 (en) | 2005-05-26 |
DE10319637A1 (en) | 2004-12-02 |
KR20060119702A (en) | 2006-11-24 |
EP1623357A2 (en) | 2006-02-08 |
JP2007524140A (en) | 2007-08-23 |
US7584238B2 (en) | 2009-09-01 |
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